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<p>In this article, it is proved that the spatial numerical range <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_843_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">\(V_A(f)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>V</mi> <mi>A</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of each element <i>f</i> of the Beurling algebra <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_843_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="64" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^1(G, \omega )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo>,</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is always convex. In fact, it is proved that the open disc <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_843_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="78" /> </InlineMediaObject> <EquationSource Format="TEX">\(U(0, \Vert f\Vert _{\omega })\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>U</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mo stretchy="false">‖</mo> <mi>f</mi> <msub> <mo stretchy="false">‖</mo> <mi>ω</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is contained in <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_843_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">\(V_A(f)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>V</mi> <mi>A</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, while the closed disc <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_843_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="79" /> </InlineMediaObject> <EquationSource Format="TEX">\(D(0, \Vert f\Vert _{\omega })\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>D</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mo stretchy="false">‖</mo> <mi>f</mi> <msub> <mo stretchy="false">‖</mo> <mi>ω</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> contains <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_843_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">\(V_A(f)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>V</mi> <mi>A</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.</p>

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The Convexity of the Spatial Numerical range in the Beurling algebra \(L^1(G, \omega )\)

  • 期刊论文

  • 发表日期：

    2025年7月27日

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  • A. B. Patel

摘要

In this article, it is proved that the spatial numerical range \(V_A(f)\) V A ( f ) of each element f of the Beurling algebra \(L^1(G, \omega )\) L 1 ( G , ω ) is always convex. In fact, it is proved that the open disc \(U(0, \Vert f\Vert _{\omega })\) U ( 0 , ‖ f ‖ ω ) is contained in \(V_A(f)\) V A ( f ) , while the closed disc \(D(0, \Vert f\Vert _{\omega })\) D ( 0 , ‖ f ‖ ω ) contains \(V_A(f)\) V A ( f ) .